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A unified theory is presented to characterize least-squares adaptive filters, in either lattice or transversal-filter form, for nonstationary processes. The derivations are based upon a geometric formulation of least-squares estimation and on the concept of displacement rank. A few basic geometric relations are shown to underlie the various algorithms. Insights into the fundamental concepts that unify lattice- and transversal-filter approaches to least-squares adaptive filters are also given. The general results are illustrated by applications to the so-called "pre-windowed" and "growing-memory covariance" formulations of the deterministic least-squares problem.